reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;

theorem
  for T being 1-sorted, S being non empty 1-sorted, f being Function of
  T, S, H being Subset-Family of S holds ("f).:H is Subset-Family of T
proof
  let T be 1-sorted, S be non empty 1-sorted, f be Function of T, S, H be
  Subset-Family of S;
  ("f).:H c= rng "f & rng "f c= bool dom f by FUNCT_3:22,RELAT_1:111;
  then ("f).:H c= bool dom f;
  hence thesis by FUNCT_2:def 1;
end;
